If you are on your own right now, don't do it yet. As one man team, you'd probably only have an hour a day to deal with it and people will find it annoying,as they tend to expect quick response when it comes to chat. Last year I was checking on this product(saas of sorts).It turns out it's just one guy behind it,who does it all. He had a calendar plugin on his website,where you can select a meeting time based on his availability.Worked like a charm. If your product is highly transactional,you may want a different approach then.

I'm curious too. I just let most of my chat messages go to email.

How do you hire for this, especially when the message volume is low (eg. average 1 per 10 minutes or so)

If you let messages go to email, do they come through with an email address you can respond to? I'd imagine that email replies that begin "I'm the founder of X" probably get a better-than-average open rate.

At the same time, I wouldn't want to frustrate people who typed something into what appeared to be a live chat window — expecting a live human — and got either a bot or a delayed email response (after a request for email address, of course). Seems like there's definitely upside potential here, but also the risk of frustrating/angering people. I'd hope there's a way to hide the chat box during certain hours, to mitigate the latter.

the user have to reenter their email, so it's opt in.

some people do expect a live response, but some also expect a bot. People have come to associate chat bubbles with chat bots I think